Spectral analysis and Riesz basis property for vibrating systems with damping

Abstract

In this thesis, we study one-dimensional wave and Euler-Bernoulli beam equations with Kelvin-Voigt damping, and one-dimensional wave equation with Boltzmann damping. The spectral property of equations with clamped boundary conditions and internal Kelvin-Voigt damping are considered. Under some assumptions on the coe±cients, it is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of ¯nite algebraic multiplicity, and the continuous spectrum that is identical to the essential spec- trum is an interval on the left real axis. The asymptotic behavior of eigenvalues is also presented. Two di®erent Boltzmann integrals that represent the memory of materials are consid- ered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the in¯nity, the spectrum of system contains a left half complex plane, which is sharp contrast to most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the later case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: There is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded.